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Phase modulation
Modulation of the phase of a signal according to the amplitude of a modulation signal. An increase in the modulation signal causes the phase of the "carrier" signal (the signal being modulated) to be advanced; in effect, it moves the signal forward in time. When the modulation signal decreases, it moves the carrier signal backwards in time. This creates distortions in the carrier signal when the modulation signal is changing. Because the ear is not sensitive to the absolute phase of a signal, phase modulation at subsonic frequencies is not very interesting unless the phase modulated signal is combined with an un-modulated signal; when this is done, phase shifting effects result. When the modulation is at audio frequences, however, it behaves somewhat like frequency modulation. When the modulation signal is increasing, the time-advancing effect that it has on the carrier signal is rather like speeding up a tape playback; it increases the frequency of the carrier signal. Similarly, a decreasing modulation signal has a time-retarding effect on the carrier, and decreases its frequency in the manner of slowing a tape playback down. However, these effects only last as long as the modulation signal is changing; at any moment when the modulation signal is constant at any value, the carrier passes through unaltered. Math ahead!... Phase modulation is mathematically related to frequency modulation as: If frequency modulation is expressed as: F = F0 + m(t) where "F0" is the base frequency of the carrier, m(t) is the amplitude of the modulation signal at time t, and F is the resulting frequency, then phase modulation using the same signals is expressed as: F = F0 + d/dt m(t) where "d/dt m(t)" is the first derivative of the modulation signal with respect to time t; or, in graphical terms, the slope of the modulation at time t. Among other things, this mathematical relationship has a consequence that John Chowning described in his original FM patents, and Yamaha exploited in the DX-7: d/dt sin(t) = cos(t) However, sin(t) and cos(t) are actually the same waveform, just with an initial phase difference; they are indistinguishable in Fourier analysis terms. So, as long as the modulation signal is a sine wave, frequency modulation and phase modulation (with audio-frequency modulation signals) produce identical results. As it turns out, in the digital context, phase modulation is easier to compute than frequency modulation. In the early 1980s, when the DX-7 was designed, microprocessors avaialble at the time didn't have enough horsepower to compute true frequency modulation in real time, but they could do phase modulation. This is why the operator waveforms on the DX-7 and the other early Yamaha FM synths are all sine waves -- because the DX-7 didn't actually compute frequency modulation internally; it computed phase modulation! But, by restricting the modulation to sine waves, it produced identical results. Yamaha was only able to offer other modulation waveforms on later models, when avaialble processors reached the point where they could compute true frequency modulation internally. Category:Math concepts Category:Synthesis methods